Chapter 7 Rational Functions - College of the Redwoods

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652 Chapter 7 Rational Functions y 10 x 10 x x − 2 y = (x + 1)(x − 4) −2 −2/3 0 1/2 3 −1/4 5 1/2 x = −1 x = 4 Figure 13. Additional points help determine the behavior near the vertical asymptote. (a) (b) (c) Figure 14. Examining end-behavior as x approaches positive infinity. If you examine the y-values in Figure 14(c), you see that they are heading towards zero (1e-4 means 1 × 10 −4 , which equals 0.0001). This implies that the line y = 0 (the x-axis) is acting as a horizontal asymptote. You can also determine the end-behavior as x approaches negative infinity (decreases without bound), as shown in the sequence in Figure 15. The result in Figure 15(c) provides clear evidence that the y-values approach zero as x goes to negative infinity. Again, this makes y = 0 a horizontal asymptote. (a) (b) (c) Figure 15. Examining end-behavior as x approaches negative infinity. Add the horizontal asymptote y = 0 to the image in Figure 13. Step 7: We can use all the information gathered to date to draw the image shown in Figure 16. Version: Fall 2007
Section 7.3 Graphing Rational Functions 653 y 10 y=0 (2,0) x 10 x=−1 x=4 Figure 16. The completed graph runs up against vertical and horizontal asymptotes and crosses the x-axis at the zero of the function. Step 8: As stated above, there are no “holes” in the graph of f. Step 9: Use your graphing calculator to check the validity of your result. Note how the graphing calculator handles the graph of this rational function in the sequence in Figure 17. The image in Figure 17(c) is nowhere near the quality of the image we have in Figure 16, but there is enough there to intuit the actual graph if you prepare properly in advance (zeros, vertical asymptotes, end-behavior analysis, etc.). (a) (b) (c) Figure 17. The user of the graphing calculator must decipher the image in the calculator’s view screen. Version: Fall 2007
Page 1 and 2: 7 Rational Functions In this chapte
Page 3 and 4: Section 7.1 Introducing Rational Fu
Page 19 and 20: Section 7.2 Reducing Rational Funct
Page 39 and 40: Section 7.3 Graphing Rational Funct
Page 51: Section 7.3 Graphing Rational Funct
Page 61 and 62: Section 7.4 Products and Quotients
Page 79 and 80: Section 7.5 Sums and Differences of
Page 95 and 96: Section 7.6 Complex Fractions 695 7
Page 97 and 98: Section 7.6 Complex Fractions 697 (
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Section 7.6 Complex Fractions 703 7
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Section 7.7 Solving Rational Equati
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Section 7.8 Applications of Rationa
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Section 7.9 Index 747 7.9 Index a a
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